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Find the coefficient of the
x^(2) term in the binomial expansion of
(4x+2)^(4).

Find the coefficient of the $x^{2}$ term in the binomial expansion of $(4 x+2)^{4}$ .

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Pascal's triangle and the Binomial Theorem

Full solution

Q. Find the coefficient of the

x^{2}

term in the binomial expansion of

(4 x+2)^{4}

.

Use Binomial Theorem: To find the coefficient of the $x^2$ term in the binomial expansion of $(4x+2)^4$ , we will use the Binomial Theorem, which states that $(a+b)^n$ expands to a sum of terms in the form of $C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient ＂ $n$ choose $k$ ＂. We are looking for the term where the power of $x$ is $2$ , which means we need the term where $k = 2$ .
Calculate Binomial Coefficient: The binomial coefficient $C(n, k)$ for the term where $k = 2$ is $C(4, 2)$ . This can be calculated as $\frac{4!}{2!(4-2)!} = \frac{(4 \times 3 \times 2 \times 1)}{(2 \times 1 \times 2 \times 1)} = 6$ .
Find the Term: Now we need to find the actual term. The term with $x^2$ will have a power of $2$ for $x$ and a power of $2$ for the constant term. Therefore, the term is $C(4, 2) \times (4x)^2 \times 2^2$ .
Calculate the Term: Calculating the term: $6 \times (4x)^2 \times 2^2 = 6 \times 16x^2 \times 4 = 6 \times 64x^2 = 384x^2$ .
Determine Coefficient: The coefficient of the $x^2$ term is therefore $384$ .

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